Computational Galerkin Methods

In the wake of the computer revolution, a large number of apparently uncon­ nected computational techniques have emerged. Also, particular methods have assumed prominent positions in certain areas of application. Finite element methods, for example, are used almost exclusively for solving structural problems; spectral methods are becoming the preferred approach to global atmospheric modelling and weather prediction; and the use of finite difference methods is nearly universal in predicting the flow around aircraft wings and fuselages. These apparently unrelated techniques are firmly entrenched in computer codes used every day by practicing scientists and engineers. Many of these scientists and engineers have been drawn into the computational area without the benefit offormal computational training. Often the formal computational training we do provide reinforces the arbitrary divisions between the various computational methods available. One of the purposes of this monograph is to show that many computational techniques are, indeed, closely related. The Galerkin formulation, which is being used in many subject areas, provides the connection. Within the Galerkin frame-work we can generate finite element, finite difference, and spectral methods.

1. Traditional Galerkin Methods.- 1.1 Introduction.- 1.2 Simple Examples.- 1.2.1 An ordinary differential equation.- 1.2.2 An eigenvalue problem.- 1.2.3 Viscous flow in a channel.- 1.2.4 Unsteady heat conduction.- 1.2.5 Burgers’ equation.- 1.3 Method of Weighted Residuals.- 1.3.1 Subdomain method.- 1.3.2 Collocation method.- 1.3.3 Least-squares method.- 1.3.4 Method of moments.- 1.3.5 Galerkin method.- 1.3.6 Generalized Galerkin method.- 1.3.7 Comparison of weighted-residual methods.- 1.4 Connection with Other Methods.- 1.5 Theoretical Properties.- 1.6 Applications.- 1.6.1 Natural convection in a rectangular slot.- 1.6.2 Hydrodynamic stability.- 1.6.3 Acoustic transmission in ducts.- 1.6.4 Flow around inclined airfoils.- 1.6.5 Microstrip disc problem.- 1.6.6 Other applications of traditional Galerkin methods.- 1.7 Closure.- References.- 2. Computational Galerkin Methods.- 2.1 Limitations of the Traditional Galerkin Method.- 2.2 Solution for Nodal Unknowns.- 2.3 Use of Low-order Test and Trial Functions.- 2.4 Use of Finite Elements to Handle Complex Geometry.- 2.5 Use of Orthogonal Test and Trial Functions.- 2.6 Evaluation of Nonlinear Terms in Physical Space.- 2.7 Advantages of Computational Galerkin Methods.- 2.8 Closure.- References.- 3. Galerkin Finite-Element Methods.- 3.1 Trial Functions and Finite Elements.- 3.1.1 One-dimensional elements.- 3.1.2 Rectangular elements in two and three dimensions.- 3.1.3 Triangular elements.- 3.2 Examples.- 3.2.1 A simplified Sturm-Liouville equation.- 3.2.2 Viscous flow in a channel.- 3.2.3 Inviscid, incompressible flow.- 3.2.4 Unsteady heat conduction.- 3.2.5 Burgers’ equation.- 3.3 Connection with Finite-Difference Formulae.- 3.4 Theoretical Properties.- 3.4.1 Convergence.- 3.4.2 Error estimates.- 3.4.3 Optimal error estimates and superconvergence.- 3.4.4 Numerical convergence results.- 3.5 Applications.- 3.5.1 Convective heat transfer.- 3.5.2 Viscous incompressible flow.- 3.5.3 Jet-flap flows.- 3.5.4 Acoustic transmission in ducts.- 3.5.5 Tidal flows.- 3.5.6 Weather forecasting.- 3.6 Closure.- References.- 4. Advanced Galerkin Finite-Element Techniques.- 4.1 Time Splitting.- 4.1.1 Thermal entry problem.- 4.1.2 Viscous compressible flow.- 4.2 Least-squares Residual Fitting.- 4.3 Special Trial Functions.- 4.3.1 Singularities.- 4.3.2 Near-wall turbulent flows.- 4.3.3 Dorodnitsyn boundary-layer formulation.- 4.4 Integral Equations.- 4.4.1 Boundary-element method.- 4.5 Closure.- References.- 5. Spectral Methods.- 5.1 Choice of Trial Functions.- 5.2 Examples.- 5.2.1 Unsteady heat conduction.- 5.2.2 Burgers’ equation.- 5.3 Techniques for Improved Efficiency.- 5.3.1 Recurrence relations.- 5.3.2 Nonlinear terms.- 5.3.3 Time differencing.- 5.4 Alternative Spectral Methods.- 5.4.1 Pseudo spectral method.- 5.4.2 The tau method.- 5.5 Orthonormal Method of Integral Relations.- 5.6 Applications.- 5.6.1 Global atmospheric modeling.- 5.6.2 Direct turbulence simulation.- 5.6.3 Other spectral applications.- 5.7 Closure.- References.- 6. Comparison of Finite Difference, Finite Element and Spectral Methods.- 6.1 Problems and Partial Differential Equations.- 6.2 Boundary Conditions and Complex Boundary Geometry.- 6.3 Computational Efficiency.- 6.4 Ease of Coding and Flexibility.- 6.5 Test Cases.- 6.5.1 Burgers’ equation.- 6.5.2 Model parabolic equations.- 6.5.3 Passive scalar convection.- 6.5.4 Open ocean modeling.- 6.6 Closure.- References.- 7. Generalized Galerkin Methods.- 7.1 A Motivation.- 7.2 Theoretical Background.- 7.2.1 Petrov-Galerkin formulation.- 7.2.2 Construction of the test function, ?k.- 7.3 Steady Convection-Diffusion Problems.- 7.3.1 Higher-order one-dimensional formulations.- 7.3.2 Two-dimensional formulations.- 7.4 Parabolic Problems.- 7.4.1 Transient convection-diffusion equation.- 7.4.2 Burgers’ equation.- 7.5 Hyperbolic Problems.- 7.6 Closure.- References.- Appendix 1 Program BURG1.- Appendix 2 Program BURG4.